Abstract: These notes give a mathematical introduction to two seemingly unrelated
topics: (i) quantum spin systems and their cycle and loop representations, due
to Tóth and Aizenman-Nachtergaele; (ii) coagulation-fragmentation stochastic
processes. These topics are nonetheless related, as we argue that the lengths
of cycles and loops satisfy an effective coagulation-fragmentation process.
This suggests that their joint distribution is Poisson-Dirichlet. These ideas
are far from being proved, but they are backed by several rigorous results,
notably of Dyson-Lieb-Simon and Schramm.